A372975 -id:A372975 - cp's OEIS frontend

A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

1, 2, 15, 4, 21, 5, 6, 7, 10, 3, 14, 9, 20, 11, 12, 13, 18, 17, 22, 19, 24, 23, 26, 25, 28, 27, 34, 29, 30, 31, 33, 8, 35, 16, 39, 32, 45, 37, 36, 41, 38, 43, 40, 47, 42, 53, 44, 49, 46, 59, 48, 61, 50, 67, 51, 64, 55, 71, 52, 73, 54, 79, 56, 81, 58, 83, 57, 70, 69, 89, 60, 77, 78, 85, 66, 65, 84

Offset: 1

Scott R. Shannon, May 26 2024

nonn

The fixed points show an unusual pattern; they begin 1, 2, 4, 69, 190, 438, 545, 725, 732, 909 and it appears, based on a graph of the sequence (see the attached image of the first 5000 terms) there may be no more. However more exist at 324388, 330574, 333069, 333531,..., 369752. Then once again there is a large gap until 2704713, 2726054, 2760963, ... . It is unclear what causes this behavior.

The sequence is conjectured to be a permutation of the positive integers.

			a(3) = 15 as a(2) = 2 and omega(2) = A001221(2) = 1, and 15 is coprime to 2 while omega(15) = A001221(15) = 2 which does not equal 1. No smaller number satisfies both of these requirements.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 terms. The green line is a(n) = n.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with purple indicating powerful numbers that are not prime powers.

Cf. A001221, A027748, A372975, A352588, A109465.

nn = 120; c[_] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; u = 3;
Do[k = u;
  While[Or[GCD[j, k] > 1, PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, May 28 2024 *)

A373350 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

Original entry on oeis.org

1, 2, 6, 30, 3, 10, 42, 4, 12, 60, 5, 14, 66, 7, 15, 70, 8, 18, 78, 9, 20, 84, 11, 21, 90, 13, 22, 102, 16, 24, 105, 17, 26, 110, 19, 28, 114, 23, 33, 120, 25, 34, 126, 27, 35, 130, 29, 36, 132, 31, 38, 138, 32, 39, 140, 37, 40, 150, 41, 44, 154, 43, 45, 156, 47, 46, 165, 49, 48, 168, 53, 50, 170, 59

Offset: 1

Scott R. Shannon, Jun 01 2024

nonn

The terms of the sequence are initially concentrated along three lines, each line containing terms with either one, two, or three-or-more distinct prime factors. See the attached images. The two lower lines, containing terms with one and two distinct prime factors, cross near n = 66, and then the upper two, containing terms with one and three-or-more distinct prime factors, cross near n = 326. At this point the upper line changes and contains terms with one distinct prime factor but also terms with more than three distinct prime factors. After a large gap the lower two lines, containing terms with two and three distinct prime factors, cross near n = 25308, and then finally, after an even longer gap, the upper two lines, containing all terms except those with three distinct prime factors, merge and become one line near n = 344310. It is likely all subsequent terms fall into one of the two remaining lines, although this is unknown.
A number with one distinct prime factor appears as a term for all n with n mod 3 = 2 up until n = 350, when a(350) = 510 = 2*3*5*17 is smaller than the next unused number with one distinct prime factor. Likewise a number with three distinct prime factors appears as a term for all n with n mod 3 = 1 up until n = 100 when a(100) = 210 = 2*3*5*7 is smaller than the next unused number with three distinct prime factors. However, rather surprisingly, a number with two distinct prime factors appears as a term for all n with n mod 3 = 0 all the way up until n = 344307, when a(344307) = 373215 = 3*5*139*179 is smaller than the next unused number with two distinct prime factors. This value of n corresponds to the start of the merging of the upper two lines near n = 344310 described above.
The fixed points begin 1, 2, 15, 18, 125, 137, 2737, 120051; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(4) = 30 as a(2) = 2 has one distinct prime factor and a(3) = 6 has two distinct prime factors, and 30 is the smallest unused number with three distinct prime factors.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500 terms. In this and other images terms with one, two, three, or more than three distinct prime factors are shown in red, yellow, green, violet respectively. The white line is a(n) = n.
Scott R. Shannon, Image of the first 50000 terms.
Scott R. Shannon, Image of the first 400000 terms.

Cf. A001221, A027748, A109465, A372975.

A373357 a(1) = 1, a(2) = 2, a(3) = 15; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1), shares a factor with a(n-2), while omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

Original entry on oeis.org

1, 2, 15, 154, 3, 10, 231, 4, 21, 110, 7, 6, 385, 8, 33, 70, 9, 14, 165, 16, 35, 66, 5, 12, 455, 27, 20, 273, 25, 18, 595, 32, 45, 182, 81, 22, 105, 11, 24, 715, 64, 39, 140, 13, 28, 195, 49, 26, 315, 128, 51, 130, 17, 36, 935, 243, 34, 285, 256, 55, 42, 121, 38, 429, 19, 44, 399, 512, 57, 170

Offset: 1

Scott R. Shannon, Jun 02 2024

nonn

The sequence uses the same rules for selecting the next term as the Yellowstone permutation A098550 but with the additional restriction that the number of distinct prime factors of a(n) must be different to both a(n-1) and a(n-2). The terms show complicated behavior, being concentrated along various curved and straight lines some of which cross and some of which only have points for various ranges of n. See the attached images.

The fixed points begin 1, 2, 32, 51, although it is possible more exist. The sequence is likely to be a permutation of the positive integers.

			a(10) = 110 as 110 shares a factor with a(8) = 4, does not share a factor with a(9) = 21, while omega(110) = 3 does not equal omega(4) = 1 or omega(21) = 2.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, where purple additionally represents powerful numbers that are not prime powers.
Scott R. Shannon, Image of the first 5000 points. Numbers with one, two, three, or four and more distinct prime factors are show as red, yellow, green and violet respectively. The white line is a(n) = n.
Scott R. Shannon, Image of the first 200000 points.

Cf. A098550, A001221, A373350, A372975, A372974.

nn = 63; c[_] := False;
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 2, 15}];
i = a[2]; j = a[3]; u = 3; v = 1; w = 2;
Do[k = u;
 While[Or[c[k],
   ! CoprimeQ[j, k],
   ! DuplicateFreeQ[{v, w, Set[x, PrimeNu[k]]}]],
 k++];
 Set[{a[n], c[k], i, j, v, w}, {k, True, j, k, w, x}];
 If[k == u, While[c[u], u++]], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 09 2024 *)

cp's OEIS Frontend

A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A373350 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A373357 a(1) = 1, a(2) = 2, a(3) = 15; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1), shares a factor with a(n-2), while omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica